Accuracy and stability of single-step methods

Above we have demonstrated the use of MATLAB solvers, with httle discussion of their performance. Here, we address these issues for the restricted class of single-step integrators 

This includes as special cases the explicit (forward) Euler method for 0 = 0, the implicit (backward) Euler method for 0 = 1, and the Crank-Nicholson method for 0 = 1/2. 

We begin first with a discussion of accuracy. Let us form Taylor series expansions around x(jk) and jc(tt+i) in the forward and reverse time directions respectively, with tk+ = tk + At, 

As the leading term in the local error is proportional to (Ar), we write 

By this analysis, we see that at every time step, there is a new local error, proportional to (Af)2, introduced into our numerical state trajectory. ... Over the course of 

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This paper reports a sensitive optical interferometric technique dual frequency modulation (DFM) for measuring and stabilizing a laser frequency by comparison, in a single step, to a radio frequency (rf) standard. Conversely, a low-noise rf source can be stabilized by a laser frequency reference. A prototype has demonstrated a resolution of 2 parts in 10 , but devices currently under development should have a resolution of 10 and an absolute accuracy of 10" . The method may be competitive with the optical frequency synthesis chain in accuracy and its simplicity suggests its convenient use in metrology, high-precision optical spectroscopy, and gravity wave detection. 

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